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Integral conditions for convergence of solutions of non-linear Robin's problem in strongly perforated domain

E. Ya. Khruslov^a, L. O. Khilkova^b, M. V. Goncharenko<sup>a

a</sup> B. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine, 47 Nauky Ave., Kharkiv 61103, Ukraine
^b Institute of Chemical Technologies of Volodymyr Dahl East Ukrainian National University, 31 Volodymyrska Str., Rubizhne 93009, Ukraine

Abstract: We consider a boundary-value problem for the Poisson equation in a strongly perforated domain $\Omega^\varepsilon =\Omega\setminus F^\varepsilon \subset R^n$ ($n\geqslant 2$) with non-linear Robin's condition on the boundary of the perforating set $F^\varepsilon$. The domain $\Omega^\varepsilon$ depends on the small parameter $\varepsilon>0$ such that the set $F^\varepsilon$ becomes more and more loosened and distributes more densely in the domain $\Omega$ as $\varepsilon\to0$. We study the asymptotic behavior of the solution $u^\varepsilon(x)$ of the problem as $\varepsilon\to0$. A homogenized equation for the main term $u(x)$ of the asymptotics of $u^\varepsilon(x)$ is constructed and the integral conditions for the convergence of $u^\varepsilon(x)$ to $u(x)$ are formulated.

Key words and phrases: homogenization, stationary diffusion, non-linear Robin's boundary condition, homogenized equation.

MSC: 35Q70

Received: 27.05.2017

Language: English

DOI: 10.15407/mag13.03.283

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